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arXiv · 0906.3595

Symétrie et théorie des groupes à travers la physique

Jacques Villain
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Abstract

Physical properties of matter are tightly related with the kind of symmetry of the medium. Group theory is a systematic tool, though not always easy to handle, to exploit symmetry properties, for instance to find the eigenvectors and eigenvalues of an operator. Certain properties (optical activity, piezoelectricity...) are forbidden in molecules or crystals of high symmetry. A few theorems (Noether, Goldstone...) establish general relations between physical properties and symmetry. Applications of group theory to condensed matter physics, elementary particle physics, quantum mechanics, electromagnetism are reviewed. Group theory is not only a tool, but also a beautiful construction which casts insight into natural phenomena. ----- Les propriétés de la matière sont liées au type de symétrie qui y règne. La théorie des groupes est un outil systématique, mais pas toujours commode, pour exploiter cette symétrie, notamment quand il faut trouver les vecteurs propres et valeurs propres d'un opérateur. Certaines propriétés (pouvoir rotatoire optique, piézoélectricité...) sont interdites dans des cristaux ou molécules de haute symétrie. Quelques théorèmes (Noether, Goldstone...) énoncent des relations générales entre les propriétés physiques et la symétrie. On passe en revue quelques applications de la théorie des groupes à la physique de la matière condensée, à la physique des particules élémentaires, à la mécanique quantique, à l'électromagnétisme. La théorie des groupes n'est pas seulement un outil, mais aussi une belle construction qui permet de mieux comprendre la nature.

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Jacques Villain. 2009-06-19. Symétrie et théorie des groupes à travers la physique. https://arxiv.org/abs/0906.3595

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